UDC 517.947.35

I. M. LAVRENT'EV

ON THE SOLVABILITY OF NONLINEAR EQUATIONS

(Presented by Academician A. N. Kolmogorov on 21 X 1966)

1. In the present note we consider the question of the existence of a solution for the nonlinear equation

\[ AF(x)=x \tag{1} \]

in a real Hilbert space. Here \(A\) and \(F(x)\) are, respectively, linear and nonlinear operators acting in \(H\). Such equations were considered in the works \((^{3,6})\). It was assumed there that the operator \(A\) is self-adjoint and bounded, and that \(F(x)\) is a potential operator satisfying certain conditions.

Here we shall consider equation (1) under more general assumptions concerning the operator \(A\); namely, we shall not assume that the operator \(A\) is bounded. In Theorem 4 we shall also consider the case when the operator \(F(x)\) is not potential. The theorems established here for equation (1) will be applied to the study of nonlinear Hammerstein-type integral equations, and we shall establish new assumptions for them.

2. Theorem 1. Let \(A\) be a positive self-adjoint operator defined on a dense set in the Hilbert space \(H\), and let \(f(x)\) be a weakly differentiable and weakly upper semicontinuous functional satisfying the condition

\[ f(x)\leq a_1(x,x)^\theta+a_2(x,x)^\mu+a_3, \]

where \(a_1<0,\ a_2\geq 0,\ a_3\geq 0,\ 0\leq \mu<\theta<1\).
Then the equation \(AF(x)=x\), where \(F(x)=\operatorname{grad} f(x)\), has at least one solution.

Theorem 2. Suppose that the following conditions are satisfied:

\(1^\circ\). \(A\) is a positive self-adjoint operator defined on a dense set in the Hilbert space \(H\); \(F(x)\) is a potential operator acting in \(H\).

\(2^\circ\). For some fixed element \(x_0\in R(A^{1/2})\) the inequality

\[ (DF(x_0+\tau h,h),h)\leq \gamma(h)\|h\|,\qquad 0\leq \tau\leq 1, \]

holds, where \(\gamma(h)\leq 0,\ \lim_{\|h\|\to\infty}\gamma(h)=-\infty,\ DF(x,h)\) is the Gateaux differential of the operator \(F(x)\).

\(3^\circ\). The potential of the operator \(F(x)\) is weakly upper semicontinuous.

Then the operator equation \(AF(x)=x\) has at least one solution.

Remark 1. If, instead of condition \(3^\circ\) of the theorem, one requires that for all \(x,h\in H\) the inequality \((DF(x,h),h)\leq 0\) hold, with equality possible only for \(h=0\), then the equation \(AF(x)=x\) has a unique solution.

Theorem 3. Let \(A\) be a positive self-adjoint operator defined on a dense set in \(H\); let \(F(x)\) be a potential operator satisfying, for some fixed \(x_0\in R(A^{1/2})\), the inequality

\[ (F(x_0+h)-F(x_0),h)\leq \gamma(\|h\|)\|h\|, \]

where \(\gamma(t)\leq 0,\ \lim_{t\to\infty}\gamma(t)=-\infty\).

Then, if the potential of the operator \(F(x)\) is weakly upper semicontinuous, the equation \(AF(x)=x\) has at least one solution.

Remark 2. If, in the conditions of the theorem, one additionally requires that for any \(x,h\in H\) the inequality
\[ (F(x+h)-F(x),h)\leq 0 \]
hold, with equality only for \(h=0\), then the equation \(AF(x)=x\) has a unique solution.

Theorem 4. Let \(A\) be a positive self-adjoint operator, defined on a dense set in \(H\); let \(F(x)\) be a nonlinear operator satisfying the conditions:

\(1^\circ.\)
\[ \|F(x+h)-F(x)\|\leq N(r)\|h\|,\quad x,\ x+h\in D_r, \]
where \(D_r\) is the ball of radius \(r\) with center at the point \(x=0\).

\(2^\circ.\)
\[ (F(x+h)-F(x),h)\leq m(r)\|h\|^2,\quad x,\ x+h\in D_r, \]
where \(m(t)\leq 0\),
\[ \lim_{t\to\infty} t m(t)=-\infty . \]
Then the operator equation \(AF(x)=x\) has a unique solution.

3. We shall apply the results presented here to establishing existence and uniqueness theorems for the solution of the nonlinear integral equation
\[ u(x)=\int_\Omega K(x,y)g(u(y),y)\,dy . \tag{2} \]

Here \(\Omega\) is a measurable set of finite or infinite measure in an \(m\)-dimensional Euclidean space.

Such propositions were considered in the works \((^4,^6,^7)\). It was assumed there that the integral operator generated by the kernel \(K(x,y)\) is self-adjoint and bounded.

However, when kernels of a more general type (Carleman kernels) are considered, this condition may fail. Recall that a Carleman kernel is a function \(K(x,y)\), measurable on the set \(\Omega\times\Omega\), such that almost everywhere in \(\Omega\times\Omega\)
\[ K(x,y)=K(y,x) \]
and, for almost all \(x\in\Omega\),
\[ \int_\Omega |K(x,y)|^2\,dy<\infty . \]

We shall consider equation (2), without assuming boundedness of the operator
\[ Au=\int_\Omega K(x,y)u(y)\,dy, \]
generated by the kernel \(K(x,y)\).

In what follows we shall require that \(A\) be a self-adjoint, positive operator defined on a dense set in \(H\). Examples of kernels generating unbounded integral operators are given in \((^1)\).

Our basic assumption concerning the nonlinear part \(g(u,x)\) consists in requiring continuity of the Nemytskii operator \(hu=g(x,u(x))\) from \(L^p\) into \(L^q\) \((1/p+1/q=1)\).

A necessary and sufficient condition for continuity of \(hu\) was established in \((^2)\).

The following propositions hold:

\(1^\circ.\) Let \(p=2\), and let the functional
\[ f(u)=\int_\Omega G(u(x),x)\,dx \]
be the potential of the operator \(hu\).

Suppose further that
\[ G(u,x)\leq a_1u^2+a_2(x)|u|^\alpha+a_3(x), \]
where \(a_1<0,\ 0\leq a_2(x)\in L^{2/(2-\alpha)},\ 0<\alpha<2\) and \(0\leq a_3(x)\in L^1\).

Then, if \(f(u)\) is weakly upper semicontinuous, equation (2) has at least one solution belonging to \(L^2(\Omega)\).

\(2^\circ\). Let the function \(g(u,x)\) satisfy the following conditions:

1) \(g(u,x)\le a_1u+a_2(x)|u|^\alpha+a_3(x)\), \(u\ge 0\);

2) \(g(u,x)\ge b_1u+b_2(x)|u|^\alpha+b_3(x)\), \(u<0\);

here \(a_1<0,\ b_1<0;\ 0\le a_2(x)\in L^{2/(1-\alpha)};\ 0\ge b_2(x)\in L^{2/(1-\alpha)};\ 0<\alpha<1;\ 0\le a_3(x)\in L^2;\ 0\ge b_3(x)\in L^2\)*.

Then, if the potential of the operator \(hu\) is weakly upper semicontinuous, equation (2) has at least one solution belonging to \(L^2(\Omega)\).

The proof of these propositions is based on Theorem 1.

\(3^\circ\). Let \(H\) be the function \(g(u,x)\) such that for all \(u,\ v\in(-\infty,+\infty)\) and almost all \(x\in\Omega\)
\[ (g(u+v,x)-g(u,x))v\le av^2, \]
where \(a<0\)**.

Then equation (2) has a unique solution in the space \(L^2(\Omega)\).

In the next proposition we shall assume that the measure of the set \(\Omega\) is infinite.

\(4^\circ\). Let \(A\) be a positive self-adjoint operator (bounded or unbounded) acting in the space \(L^2(\Omega)\). Suppose, moreover, that \(A\) is a bounded operator from \(L^p\) into \(L^q\)
\[ (1/p+1/q=1). \]

Assume further that \(H\) is the function \(g(u,x)\) satisfying the following conditions:

1) \(g(u,x)\le a_2(x)|u|^\alpha+a_3(x)\), \(\quad u\ge 0\),
\[ g(u,x)\ge -a_2(x)|u|^\alpha-a_3(x),\quad u<0; \]
here \(0\le a_2(x)\in L^{p/[(p-1)-\alpha]},\ 0<\alpha<1,\ 0\le a_3(x)\in L^q\).

2) The Nemytskii operator \(hu=g(u(x),x)\) is continuous from \(L^p\) into \(L^q\).

Then, if the functional \(f(u)\), which is the potential of the operator \(hu\), is weakly upper semicontinuous, the integral equation (2) has at least one solution belonging to the space \(L^p(\Omega)\).

Let us note that analogous propositions can also be established for systems of nonlinear integral equations of the form
\[ u_i(x)=\int_\Omega K_i(x,y)g_i(u_1(y),u_2(y),\ldots,u_n(y),y)\,dy, \]
\(i=1,2,\ldots,n\). In this case we additionally assume the existence of a function \(G(u_1,u_2,\ldots,u_n,x)\) such that
\[ g_i(u_1,u_2,\ldots,u_n,x)=\frac{\partial}{\partial u_i}G(u_1,u_2,\ldots,u_n,x). \]

The proof of these propositions is based on the method proposed by us in (7).

I express my gratitude to my scientific advisers V. V. Nemytskii and M. M. Vainberg for their advice and attention.

Moscow State University
named after M. V. Lomonosov

Received
1 IX 1966

REFERENCES

\(^{1}\) N. I. Akhiezer, UMN, 2, no. 5 (21), 93 (1947).
\(^{2}\) M. M. Vainberg, DAN, 92, no. 2, 213 (1953).
\(^{3}\) M. M. Vainberg, Variational methods for the study of nonlinear equations, 1956.
\(^{4}\) M. M. Vainberg, Uchen. zap. Mosk. obl. ped. inst., 77, no. 5, 131 (1959).
\(^{5}\) M. M. Vainberg, Sibirsk. matem. zhurn., 2, 2, 201 (1961).
\(^{6}\) M. Golomb, Math. Zs., 39, 45 (1934).
\(^{7}\) M. M. Lavrent’ev, DAN, 166, no. 2, 284 (1966).
\(^{8}\) Ya. L. Engel’son, Nauchn. dokl. vyssh. shkoly, no. 4, 75 (1958).

* The existence of such constants \(a_1,\ b_1,\ \alpha\) and functions \(a_i(x),\ b_i(x)\), \(i=1,2\), follows from the continuity condition for the Nemytskii operator.

** This condition, in particular, will be fulfilled if \(g(u,x)\) has a partial derivative with respect to \(u\), \(g'_u(u,x)\), which for almost all \(x\in\Omega\) satisfies the inequality \(g'_u(u,x)\le a\) \((a<0)\) and is bounded below.